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Orderable topological spaces Galik , Frank John
Abstract
Let (X , ਹ) be a topological space. If < is a total ordering on X , then (X , ਹ, <) is said to be an ordered topological space if a subbasis for ਹ is the collection of all sets of the form {x ∊ x  x < t} or [x ∊ x  t < x} where t ∊ X . The pair (X , ਹ) is said to be an orderable topological space if there exists a total ordering, < , on X such that (X , ਹ, <) is an ordered topological space. Definition: Let T be a subspace of the real line ǀR . Let Q be the union of all nontrivial components of T , both of whose end points belong to C1ıʀ(C1ıʀ(T) T). The following characterization of orderable subspaces of ǀR is due to M. E. Rudin. Theorem: Let T be a subspace of ǀR with the relativized usual topology. Then T is orderable if and only if T satisfies the following two conditions: (1) If T  Q is compact and (TQ) ก Clıʀ(Q) = Ø then either Q = Ø or T  Q = Ø (2) If I is an open interval of ıʀ and p is an end point of I and if {p} U(I ก(TQ)) is compact and {p} =Clıʀ(IกQ)ก C1ıʀ(I ก(TQ)), then p ∉ T or {p} is a component of T. This theorem enables us to prove a conjecture of I.L. Lynn, namely Corollary: if T contains no open compact sets then T is totally orderable. If T is a subspace of an arbitrary ordered topological space a generalization of the theorem can be made. The generalized theorem is stated and some examples are given.
Item Metadata
Title 
Orderable topological spaces

Creator  
Publisher 
University of British Columbia

Date Issued 
1971

Description 
Let (X , ਹ) be a topological space. If < is a total ordering on X , then (X , ਹ, <) is said to be an ordered topological space if a subbasis for ਹ is the collection of all sets of the form {x ∊ x  x < t} or [x ∊ x  t < x} where t ∊ X . The pair (X , ਹ) is said to be an orderable topological space if there exists a total ordering, < , on X such that (X , ਹ, <) is an ordered topological space.
Definition: Let T be a subspace of the real line ǀR . Let Q be the union of all nontrivial components of T , both of whose end points belong to C1ıʀ(C1ıʀ(T) T).
The following characterization of orderable subspaces of ǀR is due to M. E. Rudin.
Theorem: Let T be a subspace of ǀR with the relativized usual topology. Then T is orderable if and only if T satisfies the following two conditions:
(1) If T  Q is compact and (TQ) ก Clıʀ(Q) = Ø then either Q = Ø or T  Q = Ø
(2) If I is an open interval of ıʀ and p is an end point of I and if {p} U(I ก(TQ)) is compact and {p} =Clıʀ(IกQ)ก C1ıʀ(I ก(TQ)), then p ∉ T or {p} is a component of T.
This theorem enables us to prove a conjecture of I.L. Lynn, namely Corollary: if T contains no open compact sets then T is totally orderable.
If T is a subspace of an arbitrary ordered topological space a generalization of the theorem can be made. The generalized theorem is stated and some examples are given.

Genre  
Type  
Language 
eng

Date Available 
20110513

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080470

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.